OF SPACE. 41 



X, the area of the section will be -7r<»> ^^^ t^® fluxion of the solid 



00 



.f^'^x, of which the fluent is ^ — --a:^, and when arz:&, the content is 

 bb Ob 



lax, which is one third of the content of the whole prismatic or cylin- 



droidal solid. Hence a pyramid is one third of the circumscribing 



prism, and a cone one third of the circumscribing cylinder. 



182. Theorem. The fluxion of any solid is equal to 

 the parallelepiped, of which the base is equal to the section 

 of the solid, and the height to the fluxion of its height. 



For every part of a solid may be considered as touching some py- 

 ramidoidal solid, and having the same fluxion : and the fluxion ex- 

 pressed by a cylindroid is equal to a parallelepiped, on the same base, 

 and of the same height. 



183. Theorem. The curve surface of a sphere is 

 equal to the rectangle coatained by its verse sine and the 

 sphere's circumference. 



The fluxion of the surface is obviously equal to the rectangle con- 

 tained by the fluxion of the circumference and the circumference of 

 the circle of which the radius is the sine; it varies, therefore, as the 

 sine ; but the fluxion of the cosine or of the verse sine varies as the 

 sine, consequently the surface varies as the verse sine. Now, where 

 the tangent becomes parallel to the axis, the fluxion of the surface 

 becomes equal to the rectangle contained by the sphere's circum- 

 ference, and the fluxion of the verse sine : hence the whole surface 

 of any segment is equal to the whole rectangle contained by its verse 

 sine and the sphere's circumference; and the surface of the whole 

 sphere is four times the area of a great circle. 



184. Theorem. The content of a sphere is two thirds 

 of that of the circumscribing cylinder. 



The fluxion of the sphere is to that of the cylinder as the square of 

 the sine to the square of the radius ; or if the fluxion of the cylinder 

 be aabx, a being the radius, and a; the verse sine, that of the sphere will 

 be {2ax — xx)bxj or 2abxx — bxxx, of which the fluent is abx^ — Ijbx^ ; 

 which, when xizuy becomes ^a% while the content of the cylinder is 



